BELIEVE ME NOT!   A SKEPTIC's GUIDE
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Let's consider a couple of "limiting cases" of such solutions.
First, suppose that the linear restoring force is extremely weak
compared to the "drag" force 
i.e.^{13.7}
.
Then
and the solutions are
[i.e.
constant, plausible only if x = 0]
and
,
which gives the same sort of
damped behaviour as if there were no restoring force,
which is what we expected.
Now consider the case where the linear restoring force
is very strong and the "drag" force extremely weak 
i.e.
.
Then
and the solutions are
,
or^{13.8}
x(t) 
= 

(13.23) 



(13.24) 

= 

(13.25) 
where
.
We may then think of
as a
complex frequency^{13.9}
whose real part is
and whose imaginary part is .
What sort of situation does this describe?
It describes a weakly damped harmonic motion
in which the usual sinusoidal pattern damps away
within an "envelope" whose shape is that of
an exponential decay. A typical case is shown
in Fig. 13.5.
Figure:
Weakly damped harmonic motion.
The initial amplitude of x (whatever x is)
is x_{0}, the angular frequency is
and the damping rate is .
The cosinelike oscillation, equivalent to the
real part of
,
decays within the envelope function
.

Next: Generalization of SHM
Up: Damped Harmonic Motion
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Jess H. Brewer 
Last modified: Sun Nov 15 13:49:43 PST 2015